Solutions of the Dual Quaternion Matrix Equation

Abstract

In this paper, we investigate the solutions to the dual quaternion matrix equation

AXB=C

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$AXB = C$$\end{document}

, including the general solution, the solution containing only the standard part, and the solution containing only the dual part. First, we present the real representation matrix of the dual quaternion matrix and analyze the properties of its vector operator. Then we transform the dual quaternion matrix equation into a real linear system and give the equivalent condition for the existence of the solutions to the dual quaternion matrix equation

AXB=C

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$AXB=C$$\end{document}

, which includes the general solution, the solution containing only the standard part, and the solution containing only the dual part. Furthermore, we obtain the expressions of these solutions and the minimal

{\text{F}}^R

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textrm{F}^R$$\end{document}

-norm solutions to the dual quaternion matrix equation

AXB=C

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$AXB=C$$\end{document}

. Finally, we propose the algorithms and conduct numerical experiments to verify the validity of our method.

Keywords

Dual quaternion / Matrix equation / Real representation / Vector operator

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Funding

National Natural Science Foundation of China(62176112)

Natural Science Foundation of Shandong Province(ZR2022MA030)

Discipline with Strong Characteristics of Liaocheng University–Intelligent Science and Technology(319462208)

RIGHTS & PERMISSIONS

Shanghai University